| The split feasibility problem is a hot topics in the research field of the nonlin-ear functional analysis and it provides important theoretical basis for us to solve the problems which come from different spaces. In recent years, many scholars devoted to study the split feasibility problems.In 2010, Moudafi introduced the split common fixed point problem, which can be viewed as a generalization of the split feasibility problem and convex fea-sibility problem. The split equality fixed point problem introduced by Moudafi in 2013 is a generalization of the split common fixed point problem. In order to solve the split equality problem, Moudafi introduced alternating CQ-algorithm and relaxed alternating CQ-algorithm in [29] and [30], respectively, and obtained some weak convergence theorems. Those two algorithms involve the computation of two projections PC and PQ, while the computation of PC and PQ is difficult, and even impossible to compute, so his results have some limits. In order to overcome this weakness, Moudafi introduced simultaneous iterative methods in [31] and obtained weak convergence theorem. In fact, this method is to replace Pc and PQ by two firmly quasi-nonexpansive mappings T and U, which avoid the calculation of Pc and PQ, meanwhile Moudafi obtained the weak convergence theorem. However, the firmly quasi-nonexpansive mapping is very special and has strong restriction. Therefore, in this thesis, we establish some new iterative algorithms which don’t involve the computation of projections and extend the Moudafi’s results for firmly quasi nonexpansive mapping to some more general nonlinear operators, such as k-strictly pseudocontractive mappings and k-strictly asymptotically pseudononspreading mappings and so on. Meanwhile, we obtain strong convergence theorem under semi-compact.Although we have extended the operator to more general, we just obtained weak convergence theorem, if the mappings don’t own semi-compactness. So we should improve the algorithms and establish a new algorithm by using viscosity approximation method, so that we can get strong convergence theorem as the firmly nonexpansive mappings don’t own semi-compactness. We affirmly answered the open question proposed by Moudafi in [30].The content of this thesis is organized by four sections.Firstly, we illustrate the background and current state of split equality fixed point problems.Secondly, an iterative scheme of split equality fixed point problem for k-strictly pseudocontractive mappings is introduced in Hilbert spaces, and we obtain some weak and strong convergence theorems.Thirdly, we introduce an iterative scheme for the split equality fixed point problems of k-strictly asymptotically pseudononspreading mappings in Hilbert spaces and, and we obtain weak and strong theorems.Finally, by using viscosity approximation method, an new iterative scheme is introduced in Hilbert spaces, and the sequence generated by the iterative scheme strongly converges to a solution of split equality fixed point problems if the firmly nonexpansive mappings don’t own semi-compactness. |
